TY - JOUR
T1 - Super-Closeness Between the Ritz Projection and the Finite Element Solution for Some Elliptic Problems
AU - Liang , Ruosi
AU - Yan , Yue
AU - Chen , Wenbin
AU - Wang , Yanqiu
JO - Communications in Computational Physics
VL - 2
SP - 803
EP - 826
PY - 2020
DA - 2020/06
SN - 28
DO - http://doi.org/10.4208/cicp.OA-2019-0154
UR - https://global-sci.org/intro/article_detail/cicp/16954.html
KW - Finite element, Ritz projection, negative norm estimate, super-closeness estimate.
AB - <p style="text-align: justify;">We prove the super-closeness between the finite element solution and the
Ritz projection for some second order and fourth order elliptic equations in both the $H^1$&nbsp;and the $L^2$<sup>&nbsp;</sup>norms. For the fourth order problem, a Ciarlet-Raviart type mixed formulation is used in the analysis. The main tool in the proof is a negative norm estimate
of the Ritz projection, which requires $H^{q+1}$ regularity for second order elliptic equations. Therefore, the analysis is done on a domain Ω with smooth boundary, and hence
we only consider the pure Neumann boundary problems which can be discretized
naturally on such domains, if ignoring the effect of numerical integrals. For the fourth
order problem, our results amend the gap between the theoretical estimates and the
numerical examples in a previous work [22].</p>